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Find the limit of the sequence 22 – 2 32 – 2 42 – 2 52 – 2 - 22 32 42 52

Find the limit of the sequence 22 – 2 32 – 2 42 – 2 52 – 2 - 22 32 42 52

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Find the Limit of the Sequence:** The sequence is given by: \[ \frac{2^2 - 2}{2^2}, \quad \frac{3^2 - 2}{3^2}, \quad \frac{4^2 - 2}{4^2}, \quad \frac{5^2 - 2}{5^2}, \ldots \] **Explanation:** - The sequence is in the form \(\frac{n^2 - 2}{n^2}\), where \(n\) is a positive integer starting from 2. - For the first term, \(n = 2\): \[ \frac{2^2 - 2}{2^2} = \frac{4 - 2}{4} = \frac{2}{4} = \frac{1}{2} \] - For the second term, \(n = 3\): \[ \frac{3^2 - 2}{3^2} = \frac{9 - 2}{9} = \frac{7}{9} \] - For the third term, \(n = 4\): \[ \frac{4^2 - 2}{4^2} = \frac{16 - 2}{16} = \frac{14}{16} = \frac{7}{8} \] - For the fourth term, \(n = 5\): \[ \frac{5^2 - 2}{5^2} = \frac{25 - 2}{25} = \frac{23}{25} \] The pattern shows each term is of the form \(\frac{n^2 - 2}{n^2} = 1 - \frac{2}{n^2}\). **Limit Calculation:** \[ \lim_{n \to \infty} \left(1 - \frac{2}{n^2}\right) = 1 - 0 = 1 \] Therefore, the limit of the sequence as \(n\) approaches infinity is 1.
Transcribed Image Text:**Find the Limit of the Sequence:** The sequence is given by: \[ \frac{2^2 - 2}{2^2}, \quad \frac{3^2 - 2}{3^2}, \quad \frac{4^2 - 2}{4^2}, \quad \frac{5^2 - 2}{5^2}, \ldots \] **Explanation:** - The sequence is in the form \(\frac{n^2 - 2}{n^2}\), where \(n\) is a positive integer starting from 2. - For the first term, \(n = 2\): \[ \frac{2^2 - 2}{2^2} = \frac{4 - 2}{4} = \frac{2}{4} = \frac{1}{2} \] - For the second term, \(n = 3\): \[ \frac{3^2 - 2}{3^2} = \frac{9 - 2}{9} = \frac{7}{9} \] - For the third term, \(n = 4\): \[ \frac{4^2 - 2}{4^2} = \frac{16 - 2}{16} = \frac{14}{16} = \frac{7}{8} \] - For the fourth term, \(n = 5\): \[ \frac{5^2 - 2}{5^2} = \frac{25 - 2}{25} = \frac{23}{25} \] The pattern shows each term is of the form \(\frac{n^2 - 2}{n^2} = 1 - \frac{2}{n^2}\). **Limit Calculation:** \[ \lim_{n \to \infty} \left(1 - \frac{2}{n^2}\right) = 1 - 0 = 1 \] Therefore, the limit of the sequence as \(n\) approaches infinity is 1.
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